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The problem given is a sequence defined by:

\[
a_n = \frac{1}{2} + \frac{2}{2^2} + \frac{3}{2^3} + \dots + \frac{n}{2^n}
\]

This sequence can be written as a sum:

\[
a_n = \sum_{k=1}^{n} \frac{k}{2^k}
\]

To analyze and possibly find a closed-form expression for \(a_n\), let's break down the components.

### Step-by-Step Solution Outline

1. **Term Identification**: Each term in the series has the form \(\frac{k}{2^k}\), where \(k\) is the term index.
2. **Approach**: Such series can often be evaluated by finding a generating function or using calculus techniques if possible, but in some cases, they do not simplify nicely.
3. **Known Result**: A common result for sums involving terms like \(\frac{k}{r^k}\) (with \(r > 1\)) can sometimes be derived using known formulas for geometric or arithmetic-geometric series.

Would you like the detailed steps for finding the closed form or just an approximation of the series sum for large \(n\)?

Given the sequence defined by \(a_n = \frac{1}{2} + \frac{2}{2^2} + \frac{3}{2^3} + \dots + \frac{n}{2^n}\), find the value of \(a_n\).

Explore the sequence \(a_n = \frac{1}{2} + \frac{2}{2^2} + \dots + \frac{n}{2^n}\), with a step-by-step approach to summing arithmetic-geometric series. Ideal for students in Grades 11-12 learning advanced summation techniques.

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